Scholarlink

Abstract <p>In a rectangle <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="256" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega = \{ (x,t)\,|\,0 &lt; x &lt; 1,\;0 &lt; t &lt; T\} \)</EquationSource> <!--ComMat2470183Denisov-m1--> </InlineEquation>, we consider an initial-boundary value problem for a singularly perturbed parabolic equation </p> <p><Equation ID="Equaa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_Equaa.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="330" /> </MediaObject> <EquationSource Format="TEX">\({{\varepsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\varepsilon ),\quad (x,t) \in \Omega ,\)</EquationSource> <!--ComMat2470183Denisov-m2--> </Equation></p> <p><Equation ID="Equab"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_Equab.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="594" /> </MediaObject> <EquationSource Format="TEX">\(u(x,0,\varepsilon ) = \varphi (x),\quad 0 \leqslant x \leqslant 1,\quad u(0,t,\varepsilon ) = {{\psi }_{1}}(t),\quad u(1,t,\varepsilon ) = {{\psi }_{2}}(t),\quad 0 \leqslant t \leqslant T.\)</EquationSource> <!--ComMat2470183Denisov-m3--> </Equation></p> <p>It is assumed that, at the corner points <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\((k,0)\)</EquationSource> <!--ComMat2470183Denisov-m4--> </InlineEquation> of the rectangle <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <!--ComMat2470183Denisov-m5--> </InlineEquation>, where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(k = 0\)</EquationSource> <!--ComMat2470183Denisov-m6--> </InlineEquation> or 1, the function <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="142" /> </InlineMediaObject> <EquationSource Format="TEX">\(F(u) = F(u,k,0,0)\)</EquationSource> <!--ComMat2470183Denisov-m7--> </InlineEquation> has the form</p> <p><Equation ID="Equac"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_Equac.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="304" /> </MediaObject> <EquationSource Format="TEX">\(F(u) = {{u}^{3}} - u_{0}^{3},\quad {\text{where}}\quad {{u}_{0}} = {{u}_{0}}(k) &lt; 0.\)</EquationSource> <!--ComMat2470183Denisov-m8--> </Equation></p> <p>To construct the asymptotics of the solution, we use a nonlinear method of corner boundary functions. Previously, the case was considered when the boundary value <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_IEq9.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <!--ComMat2470183Denisov-m9--> </InlineEquation> at the corner points is separated from the inflection point <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(u = 0\)</EquationSource> <!--ComMat2470183Denisov-m10--> </InlineEquation> by the condition</p> <p><Equation ID="Equad"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_Equad.gif" Format="GIF" Height="38" Rendition="HTML" Resolution="72" Type="Linedraw" Width="193" /> </MediaObject> <EquationSource Format="TEX">\({{u}_{0}}(k) &lt; \varphi (k) \leqslant \frac{{{{u}_{0}}(k)}}{2} &lt; 0,\)</EquationSource> <!--ComMat2470183Denisov-m11--> </Equation></p> <p>in which the role of barrier functions was played by functions of the simplest type, suitable in the entire domain. In this paper, we consider the case of</p> <p><Equation ID="Equae"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_Equae.gif" Format="GIF" Height="38" Rendition="HTML" Resolution="72" Type="Linedraw" Width="132" /> </MediaObject> <EquationSource Format="TEX">\(\frac{{{{u}_{0}}(k)}}{2} &lt; \varphi (k) &lt; 0,\)</EquationSource> <!--ComMat2470183Denisov-m12--> </Equation></p> <p>in which the domain has to be divided into parts in order to construct in each subdomain its individual barrier functions taking into account their continuous matching at the common boundaries of the subdomains and then to smooth out the piecewise continuous lower and upper solutions. As a result, we obtain a complete asymptotic expansion of the solution at <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2163_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon \to 0\)</EquationSource> <!--ComMat2470183Denisov-m13--> </InlineEquation> and substantiate its uniformity in the closed rectangle.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Nonlinear Method of Corner Boundary Functions with the Influence of an Inflection Point

  • 期刊论文

  • 发表日期：

    2025年1月1日

查看全文
  • A. I. Denisov,
  • I. V. Denisov

摘要

Abstract

In a rectangle \(\Omega = \{ (x,t)\,|\,0 < x < 1,\;0 < t < T\} \) , we consider an initial-boundary value problem for a singularly perturbed parabolic equation

\({{\varepsilon }^{2}}\left( {{{a}^{2}}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} - \frac{{\partial u}}{{\partial t}}} \right) = F(u,x,t,\varepsilon ),\quad (x,t) \in \Omega ,\)

\(u(x,0,\varepsilon ) = \varphi (x),\quad 0 \leqslant x \leqslant 1,\quad u(0,t,\varepsilon ) = {{\psi }_{1}}(t),\quad u(1,t,\varepsilon ) = {{\psi }_{2}}(t),\quad 0 \leqslant t \leqslant T.\)

It is assumed that, at the corner points \((k,0)\) of the rectangle \(\Omega \) , where \(k = 0\) or 1, the function \(F(u) = F(u,k,0,0)\) has the form

\(F(u) = {{u}^{3}} - u_{0}^{3},\quad {\text{where}}\quad {{u}_{0}} = {{u}_{0}}(k) < 0.\)

To construct the asymptotics of the solution, we use a nonlinear method of corner boundary functions. Previously, the case was considered when the boundary value \(\varphi \) at the corner points is separated from the inflection point \(u = 0\) by the condition

\({{u}_{0}}(k) < \varphi (k) \leqslant \frac{{{{u}_{0}}(k)}}{2} < 0,\)

in which the role of barrier functions was played by functions of the simplest type, suitable in the entire domain. In this paper, we consider the case of

\(\frac{{{{u}_{0}}(k)}}{2} < \varphi (k) < 0,\)

in which the domain has to be divided into parts in order to construct in each subdomain its individual barrier functions taking into account their continuous matching at the common boundaries of the subdomains and then to smooth out the piecewise continuous lower and upper solutions. As a result, we obtain a complete asymptotic expansion of the solution at \(\varepsilon \to 0\) and substantiate its uniformity in the closed rectangle.